Determine if the series converges absolutely, converges conditionally, or diverges.Justify. 9)
_
Q
n=1
(-1)n
3
4
-
4
n
n
The series can be written as:
∑((-1)^n)/(3(4^n/n))
To determine if the series converges absolutely or conditionally, we will first check if it converges conditionally by ignoring the signs of the terms and checking if the series ∑(1/(3(4^n/n))) converges.
Using the ratio test:
lim(n→∞) |(1/(3(4^(n+1)/(n+1)))) / (1/(3(4^n/n)))| = lim(n→∞) |(4^n/n) / (4^(n+1)/(n+1))|
= lim(n→∞) |(4^n / n) * (n+1) / 4^(n+1)|
= lim(n→∞) |(n+1)/(4n)|
= 1/4
Since the ratio is less than 1, the series ∑(1/(3(4^n/n))) converges.
Since the series ∑((-1)^n)/(3(4^n/n)) has the same terms as ∑(1/(3(4^(n+1)/(n+1)))), the original series converges conditionally.
Therefore, the series converges conditionally.